The statement is as follows: Let be a simply connectedopen subset of the complex plane containing a finite list of points, and a function defined and holomorphic on. Let be a closed rectifiable curve in which does not meet any of the, and denote the winding number of around by. The line integral of around is equal to times the sum of residues of at the points, each counted as many times as winds around the point: If is a positively orientedsimple closed curve, if is in the interior of, and 0 if not, so with the sum over those inside. The relationship of the residue theorem to Stokes' theorem is given by the Jordan curve theorem. The general plane curve must first be reduced to a set of simple closed curves whose total is equivalent to for integration purposes; this reduces the problem to finding the integral of along a Jordan curve with interior. The requirement that be holomorphic on is equivalent to the statement that the exterior derivative on. Thus if two planar regions and of enclose the same subset of, the regions and lie entirely in, and hence is well-defined and equal to zero. Consequently, the contour integral of along is equal to the sum of a set of integrals along paths, each enclosing an arbitrarily small region around a single — the residues of at. Summing over, we recover the final expression of the contour integral in terms of the winding numbers. In order to evaluate real integrals, the residue theorem is used in the following manner: the integrand is extended to the complex plane and its residues are computed, and a part of the real axis is extended to a closed curve by attaching a half-circle in the upper or lower half-plane, forming a semicircle. The integral over this curve can then be computed using the residue theorem. Often, the half-circle part of the integral will tend towards zero as the radius of the half-circle grows, leaving only the real-axis part of the integral, the one we were originally interested in.
Examples
An integral along the real axis
The integral arises in probability theory when calculating the characteristic function of the Cauchy distribution. It resists the techniques of elementary calculus but can be evaluated by expressing it as a limit of contour integrals. Suppose and define the contour that goes along the real line from to and then counterclockwise along a semicircle centered at 0 from to. Take to be greater than 1, so that the imaginary unit is enclosed within the curve. Now consider the contour integral Since is an entire function, this function has singularities only where the denominator is zero. Since, that happens only where or. Only one of those points is in the region bounded by this contour. Because is the residue of at is According to the residue theorem, then, we have The contour may be split into a straight part and a curved arc, so that and thus Using some estimations, we have and The estimate on the numerator follows since, and for complex numbers along the arc, the argument of lies between 0 and. So, Therefore, If then a similar argument with an arc that winds around rather than shows that and finally we have
The fact that has simple poles with residue 1 at each integer can be used to compute the sum Consider, for example,. Let be the rectangle that is the boundary of with positive orientation, with an integer. By the residue formula, The left-hand side goes to zero as since the integrand has order. On the other hand, Thus, the residue is. We conclude: which is a proof of the Basel problem. The same trick can be used to establish the sum of the Eisenstein series: We take with a non-integer and we shall show the above for. The difficulty in this case is to show the vanishing of the contour integral at infinity. We have: since the integrand is an even function and so the contributions from the contour in the left-half plane and the contour in the right cancel each other out. Thus, goes to zero as.
